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Unformatted text preview: Midterm 1 Solutions 1. (a) Find 2 2 matrices A and B such that AB 6 = BA . There are many examples to answer this problem. Here is one: Let A = 1 0 0 0 and B = 0 1 1 1 . Then AB = 0 1 0 0 and BA = 0 0 1 0 so that AB 6 = BA . (b) For general n n matrices A and B find an expression for ( A + B ) 2 . Let A and B be n n matrices. Then, ( A + B ) 2 = ( A + B )( A + B ) = A ( A + B ) + B ( A + B ) = A 2 + AB + BA + B 2 . Where the last two equalities are satisfied by distributive properties of matrix operations. (c) Write the following system of Differential Equations in matrix vector form: x 1 = 3 x 1 + 5 x 2 + sin( t ) , x 2 = x 1 + 2 x 2 + cos( t ) x 1 x 2 = 3 5 1 2 x 1 x 2 + sin( t ) cos( t ) 2. Answers without correct supporting work will not count, even if correct. Let A = 1 1 2 2 1 11 4 3 10 . Compute A 1 using Elementary Row Transformation and check your work by computing A 1 A ....
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This note was uploaded on 03/30/2010 for the course MATH 225 taught by Professor Guralnick during the Spring '07 term at USC.
 Spring '07
 Guralnick
 Math, Matrices

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